I recently just made a felted fair isle bag and the inside of the bag does felt a tiny bit but not enough for the strands that are carried across to 'stick' to each other. In other words, you can still stick a finger under the majority of the strands.
My friend who sews is lining it for me since the loose strands will drive me nuts if they get caught on things that I put inside the bag.
As HTH said, it is critical that you shape the bag each time you take it out of the washer. I put mine through the wash cycle three times and after each time the cycle ended I shaped it before throwing it back in paying special attention to shape the corners and the length to width ratio of the bag.